Optimal retailers ordering policies unde
Int. J. Industrial and Systems Engineering, Vol. 10, No. 3, 2012
Optimal retailer’s ordering policies under two-stage
partial trade credit financing in a supply chain
Chandra K. Jaggi*, K.K. Aggarwal and
Mona Verma
Department of Operational Research,
Faculty of Mathematical Sciences,
New Academic Block,
University of Delhi,
Delhi 110007, India
Fax: +91 11 27666672
E-mail: [email protected]
E-mail: [email protected]
*Corresponding author
Abstract: In this paper, an attempt has been made to investigate partial trade
credit financing for two levels of supply chain, i.e. the retailer as well as the
customer must make a partial payment initially in order to make them eligible
for availing the delay period for the rest of their purchases. We have developed
the retailer’s inventory system as a cost minimisation problem to determine the
retailer’s optimal ordering policies. Further, we have deduced the models
presented by Huang (2005, 2003) and Goyal (1985). It has been established
numerically that offering partial trade credit in two-stage supply chain is
beneficial as per retailer’s perspective. Comprehensive sensitive analysis along
with a case study has also been presented.
Keywords: partial trade credit; EOQ; economic order quantity; inventory;
supply chain.
Reference to this paper should be made as follows: Jaggi, C.K.,
Aggarwal, K.K. and Verma, M. (2012) ‘Optimal retailer’s ordering policies
under two-stage partial trade credit financing in a supply chain’, Int. J.
Industrial and Systems Engineering, Vol. 10, No, 3, pp.277–299.
Biographical notes: Chandra K. Jaggi is an Associate Professor in the
Department of Operational Research, University of Delhi, India. He earned his
PhD and MPhil in Inventory Management and Masters in Operational Research
from Department of Operational Research, University of Delhi. His area of
teaching includes inventory and financial management. His area of research is
production/inventory and supply chain management. He has publications in Int.
J. Production Economics, Journal of Operational Research Society, European
Journal of Operational Research, Int. J. Systems Sciences, Int. J. Procurement
Management, TOP, Int. J. Applied Decision Sciences, OPSEARCH,
Investigacion Operacional Journal, Int. J. Operational Research, Advanced
Modeling and Optimization, Canadian Journal of Pure and Applied Sciences,
Journal of Information and Optimization Sciences, Int. Journal of
Mathematical Sciences, Indian Journal of Mathematics and Mathematical
Sciences, Indian Journal of Management and Systems, etc.
Copyright © 2012 Inderscience Enterprises Ltd.
277
278
C.K. Jaggi, K.K. Aggarwal and M. Verma
K.K. Aggarwal is an Assistant Professor in the Department of Operational
Research, University of Delhi, India. He obtained his PhD in Inventory
Management and Masters in Operational Research from Department of
Operational Research, University of Delhi. Prior to his appointment as
Assistant Professor, he served as Scientist in National Informatics Centre
(India). His research interests and teaching include inventory modelling,
financial engineering and network analysis. He has published more than 15
research papers in Int. J. Production Economics, Indian Journal of
Mathematics and Mathematical Sciences and Investigacion Operacional
Journal.
Mona Verma is a Research Scholar in the Department of Operational Research,
Faculty of Mathematical Sciences, University of Delhi, India. She has
completed her MPhil in Inventory Management and Masters in Operational
Research from the Department of Operational Research, University of Delhi.
Currently, she is pursuing her PhD in Operational Research. Her research
interest lies in inventory management and supply chain management.
1
Introduction
In today’s business transactions, it is generally observed that the retailers are allowed a
fixed time period before they settle their account with the supplier, which is simply
known as trade credit period. During this period, the retailer can sell the goods and earn
interest on revenue generated. After this period, if the account is not settled, then the
supplier charges interest on the remaining inventory as per the terms and conditions
agreed upon. This is termed as one-level trade credit financing, which has been explored
by many researchers at length. Initially, Goyal (1985) had analysed the effects of trade
credit on the optimal inventory policy. Chung (1998) developed an alternative approach
to determine the EOQ under the condition of permissible delay in payments. Further,
Aggarwal and Jaggi (1995) considered the inventory model with an exponential
deterioration rate under the condition of permissible delay in payments. Jamal et al.
(1997) and Chang and Dye (2001) further generalised the model with shortages. Further,
Hwang and Shinn (1997), Jamal et al. (2000), Chung et al. (2005), Chung and Liao
(2006) also presented their work under varying assumptions. Abad and Jaggi (2003) first
offered a supplier–retailer integrated model following a lot-for-lot shipment policy under
permissible delay in payment. In their model, the retailer takes advantage of the trade
credit offered by the supplier but requires his/her customers to pay for the items
immediately. In all the above papers, researchers have assumed that the retailer does not
offer credit period to his/her customer. Huang (2003) presented an inventory model
assuming that the retailer also offers a credit period to his/her customer which is shorter
than the credit period offered by the supplier to him, in order to stimulate his own
demand. This type of trade credit is termed as two-level trade credit policy in a supply
chain modelling. Later on, Huang (2007) investigated the optimal retailer’s replenishment
decisions with two levels of trade credit policy in the EPQ framework. Ho et al. (2007)
also developed an integrated supplier retailer inventory model with demand rate increases
exponentially with the customer’s credit period. Further, Jaggi et al. (2008) incorporated
the concept of credit-linked demand and developed an inventory model under two levels
of trade credit policy to determine the optimal credit as well as replenishment policy
Optimal retailer’s ordering policies
279
jointly, for the retailer. Another realistic phenomenon of the supply chain modelling is
partial trade credit, i.e. the retailer/customer need to make some partial amount for their
purchases in order to make themselves eligible for availing the credit period for the rest
of the amount .Huang (2005) developed optimal retailer’s policy for one level where the
supplier offers partial trade credit to the retailer. Adding to this, Huang and Hsu (2008)
expanded the model where retailer gets full trade credit but offers partial trade credit to
his/her customer. Thangam and Uthayakumar (2008) developed the partial trade credit
financing in an EPQ model under the same environment.
Unfortunately, none of the researchers have incorporated the partial trade credit for
supplier as well as retailer. In this paper, an attempt has been made to investigate partial
trade credit financing for a two level of supply chain, i.e. the retailer as well as the
customer must make a partial payment initially in order to make them eligible for
availing the delay period for the rest of their purchases. We have developed the retailer’s
inventory system as a cost minimisation problem to determine the retailer’s optimal
ordering policies. Further, we have deduced the models presented by Huang (2003, 2005)
and Goyal (1985). It has been established numerically that offering partial trade credit in
two-stage supply chain is beneficial from the retailer’s perspective.
2
Assumptions and notations
We formulate a two-echelon supply chain model, for the general case, where the supplier
as well as the retailer offers partial trade credit to their downstream mentor. The case of
full credit then is just the special case of the general model. The nomenclature is as
follows:
D demand rate per year
A
ordering cost per order
c
unit purchase price for retailer
p
unit selling price for retailer, c d p
h
unit stockholding cost per year excluding interest charges
I e interest earned per $ per year for retailer
I p interest charged per $ in stocks per year for retailer
M retailer’s credit period offered by the supplier for settling the accounts
N customer’s credit period offered by the retailer for settling the account
D
the percentage of permissible delay in payments for retailer, 0 d D d 1
E
the percentage of permissible delay in payments for customer, 0 d E d 1
T
cycle time in years
TRC(T ) annual total relevant cost, which is a function of T .
280
C.K. Jaggi, K.K. Aggarwal and M. Verma
The following assumptions are made in the model:
1
Demand rate is known and constant.
2
Shortages are not allowed.
3
Time horizon is infinite.
4
The supplier offers a partial trade credit period M to the retailer. As the order is
received, the retailer has to make a partial payment of (1 D )cDT to the supplier.
During this period, the retailer earns interest
I e (d, t, I p ) on the revenue
generated from sales at the selling price p ! c, whereas if the account is not settled
at M , then the supplier charges interest I p on the remaining stock at the purchase
price c.
5
The retailer also offers a partial trade credit period N to his/her customer, i.e. his/her
customer must make a partial payment E pDT to the retailer, and they must pay off
the rest of the amount at the end of the trade credit period N (independent of M ),
which is offered to him by the retailer. Now the retailer earns interest I e on the
partial payment, which he receives from his/her customer.
3
The retailer’s problem
The retailer wants to stimulate his sales by offering partial trade credit E to his/her
customer, while making himself eligible for the same by giving an initial payment on
(1 D ) units to the supplier .The objective is to minimise annual total cost, which
comprises of the following elements:
Annual purchasing cost = cD
Annual ordering cost = A/T
Annual stockholding cost (excluding interest charges) = DTh/2
The annual total relevant cost for the retailer can be expressed as shown below:
TRC(T) = Annual purchase cost + Annual ordering cost + Annual stockholding
cost + Annual interest payable – Annual interest charged
After the credit period M , the retailer has to make payment on the items in stock at
the interest I p and he will earn interest at the rate I e if the customer does not pay for the
items at the specified credit period N offered by him. The two situations that may arise
are as follows:
1
M tN
2
M N
Case 1 M t N : According to assumption 4 and 5, there are four subcases to occur for
interest payable per year and interest earned per year for the retailer.
1
N dM d
M
d T (Figure 1)
1D
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Optimal retailer’s ordering policies
In this case, since the retailer makes a partial payment of (1 D ) units to the supplier, the
interest paid, therefore, will be on the balance amount of purchases made by him.
Annual interest payable
Annual interest earned
ª DT 2 / 2 D DTM º
»
cI p «
«
»
T
¬
¼
ª E DN 2 / 2 (( DN DM )( M N ) / 2)) º
»
pI e «
«
»
T
¬
¼
Therefore, the annual total relevant cost will be
TRC1 (T )
2
M dT d
cD
ª
M 2 N2
E N2
A DTh
ªT
º
cI p D « D M » pI e D «
« 2T
2
2T
T
¬2
¼
¬
º»
»
¼
(1)
M
(Figure 2)
1D
Annual interest payable
ª (1 D )2 DT 2 / 2 D(T M )2 / 2 º
»
cI p «
«
»
T
¬
¼
ª (1 D ) 2 T 2 (T M )2 º
cDI p «
»
2T
¬
¼
Annual interest earned
ª E DN 2 / 2 (( DN DM )( M N )) / 2 º
»
pI e «
«
»
T
¬
¼
Thus, the annual total relevant cost is given by
TRC 2 (T )
cD
ª (1 D ) 2 T (T M ) 2 º
A DTh
cI p D «
»
T
2
2
2T
¬
¼
ª
M 2 N2
E N2
pI e D «
« 2T
2T
¬
3
º»
(2)
»
¼
N d T d M (Figure 3)
Annual interest payable
ª (1 D ) 2 DT 2 / 2 º
»
cI p «
«
»
T
¬
¼
Annual interest earned
ª E DN 2 / 2 (( DN DT )(T N )) / 2 DT ( M T ) º
»
pI e «
«
»
T
¬
¼
282
C.K. Jaggi, K.K. Aggarwal and M. Verma
Hence, the annual total relevant cost will be
TRC3 (T )
A DTh
(1 D ) 2 T
cI p D
T
2
2
2
2
ª
º
EN 2 T N
pDI e «
( M T )»
« 2T
»
2T
¬
¼
cD
4
T d N (Figure 4)
º»
Annual interest payable
ª (1 D ) 2 DT 2 / 2
cI p «
«
T
¬
Annual interest earned
ª E DT 2 / 2 ( E DT ( N T )) ( DT ( M N )) º
»
pI e «
«
»
T
¬
¼
Figure 1
N d M / (1 D ) d T
Figure 2
M d T d M / (1 D )
»
¼
(3)
Optimal retailer’s ordering policies
Figure 3
N dT d M
Figure 4
TdN
283
Consequently, the annual total relevant cost is given as
TRC4 (T )
cD
A DTh
(1 D ) 2 T
ET
ª
º
cI p D
pI e D « M
(1 E ) N »
T
2
2
2
¬
¼
(4)
From the above arguments, the annual total relevant cost for the retailer can be
expressed as
TRC(T )
°TRC1 (T )
°
°TRC (T )
2
®
°
°TRC3 (T )
°TRC (T )
¯
4
if
M
dT
1 D
M
1 D
if N d T d M
if M d T d
if T t N
(5a)
(5b)
(5c)
(5d)
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C.K. Jaggi, K.K. Aggarwal and M. Verma
Since TRC1 ( M / (1 D )) TRC2 ( M / (1 D )), TRC2 ( M ) TRC3 ( M ) and TRC3 ( N )
TRC4 ( N ), TRC(T ) is continuous and well defined. All TRCi (T ) i 1 4 and TRC(T )
are defined on T > 0. To minimise the annual total cost, we take the first order and second
order derivatives of TRCi (T ) i 1 4 with respect to T . Therefore, Equations (1)–(4)
yield
ª 2 A pI e D M 2 (1 E ) N 2
«
«
2T 2
¬
TRC1cc(T )
ª 2 A pI e D M 2 (1 E ) N 2
«
«
T3
¬
TRC 2c (T )
ª 2 A cI p DM 2 pI e D M 2 (1 E ) N 2
«
«
2T 2
¬
»
¼
«
¬
2
P
º
»
¼
(6)
º»
(7)
»
¼
ª h cI p 1 (1 D ) 2
D«
«
2
¬
ȼ
º»
»
¼
(8)
»
¼
º»
TRCcc2 (T )
ª 2 A cI p DM 2 pI e D M 2 (1 E ) N 2
«
«
T3
¬
TRC3c (T )
ª h cI p (1 D ) 2 pI e º
ª 2 A pI e DN 2 (1 E ) º
«
»
«
D
»
2
2T 2
«¬
»¼
¬«
¼»
(10)
TRC3cc (T )
ª 2 A pI e DN 2 (1 E ) º
«
»!0
T3
«¬
»¼
(11)
TRCc4 (T )
ª h cI p (1 D ) 2 pI e E º
ª Aº
»
« 2 » D«
2
¬T ¼
«¬
»¼
(12)
TRCcc4 (T )
Let [1
»º D ª h cI
TRC1c (T )
2A
T3
(9)
»
¼
!0
2 A pI e D( M 2 (1 E ) N 2 ), [2
(13)
2 A cI p DM 2 pI e D ( M 2 (1 E ) N 2 ).
Then we can find that [1 d [2 . Equations (11) and (13) imply that TRC3 (T ) and
TRC4 (T ) is convex on T > 0. However, TRC1 (T ) is convex on T > 0 if [1 ! 0
and TRC2 (T ) is convex on T > 0 if [2 ! 0. Furthermore, we have
§ M ·
TRC1c ¨
¸
© 1 D ¹
§ M ·
TRCc2 ¨
¸ , TRCc2 ( M )
©1D ¹
TRCc3 ( M ) and TRCc3 ( N )
TRCc4 ( N )
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Optimal retailer’s ordering policies
Therefore, Equation (5a)–(5d) imply that TRC(T ) is convex on T > 0 if [1 ! 0. Then we
can obtain the following results.
Theorem 1:
1
If [2 d 0, then TRC (T ) is convex on (0, M ] and concave in [ M , f).
2
If [ 1 t 0 and [2 ! 0, then TRC (T ) is convex on (0,
M
] and concave in
1D
M
[
, f ).
1 D
3
If [1 d 0, then TRC (T ) is convex on (0, f).
Case 2
M N
Annual ordering cost = A/T
Annual stockholding cost (excluding interest charges) = DTh/2
According to assumption 5 and assumption, there are two cases to occur in interest
payable per year and interest earned per year, i.e.
1
M dT
2
M tT
1
M d T (Figure 5)
Annual interest payable
Annual interest earned
2
ª (1 D ) 2 T (T M )2 º
cI p D «
»
2
2T
¬
¼
pI e
E DM 2
2T
M t T (Figure 6)
Annual interest payable
Annual interest earned
ª (1 D )2 T º
cI p D «
»
2
¬
¼
ª E DT 2
º
pI e «
E DT ( M T ) »
¬ 2
¼
Tº
ª
pI e E D « M »
2¼
¬
From the above arguments, the annual total relevant cost for the retailer can be
expressed as
286
C.K. Jaggi, K.K. Aggarwal and M. Verma
Figure 5
Interest earned when M d T
Figure 6
Interest earned when M t T
TRC(T) = Annual purchase cost + Annual ordering
cost + Annual interest payable – Annual interest charged
where
TRC(T )
cost + annual
stockholding
TRC5 (T ) if M d T
®
¯TRC6 (T ) if M t T
(14a)
(14b)
where
TRC5 (T )
cD
ª (1 D )2 T (T M ) 2 º pI e D E M 2
A DTh
cI p D «
»
2
2
2T
2T
T
¬
¼
(15)
287
Optimal retailer’s ordering policies
and
cD
TRC6 (T )
ª (1 D )2 T º
A DTh
Tº
ª
cI p D «
» pI e D E « M »
T
2
2
2¼
¬
¬
¼
(16)
Since TRC5 ( M ) TRC6 ( M ), TRC(T ) is continuous and well defined. All TRCi (T ),
i 5, 6 and TRC(T ) are defined on T > 0. Equations (15) and (16) yield
TRC5c (T )
ª h cI p 1 (1 D ) 2
ª 2 A cI p DM 2 pI e D E M 2 º
» D«
«
«
2
2T 2
«¬
»¼
¬
TRCcc5 (T )
ª 2 A cI p DM 2 pI e D E M 2 º
«
»
T3
«¬
»¼
TRCc6 (T )
2
ª A º D h cI p (1 D ) pI e E
« 2 »
2
¬T ¼
TRC6cc (T )
ª2Aº
«T3 » ! 0
¬ ¼
º»
»
¼
(17)
(18)
and
(19)
(20)
Equation (20) implies that TRC6 (T ) is convex on T > 0 and Equation (18) implies that
TRC5 (T ) will be convex on T > 0 when 2 A cI p DM 2 pI e DM 2 E ! 0. Furthermore, we
have TRC5c ( M ) TRC6c ( M ). Therefore, Equation (14a) and (14b) implies that TRC(T )
is convex on T > 0 when 2 A cI p DM 2 pI e DM 2 E ! 0.
4
Determination of the optimal cycle time T *
Suppose that M t N
Let TRCci (Ti* ) 0 for all i = 1–4. We can obtain
Case 1
T1*
T2*
T3*
2 A pI e D M 2 (1 E ) N 2
D h cI p
,
if [1 ! 0, [2 ! 0
(21)
,
(22)
2 A cI p DM 2 pI e D M 2 (1 E ) N 2
D h cI p 1 (1 D ) 2
2 A pI e D(1 E ) N 2
D h cI p (1 D ) 2 pI e
if [ 2 ! 0
(23)
288
C.K. Jaggi, K.K. Aggarwal and M. Verma
T4*
2A
D h cI p (1 D )2 pI e E
(24)
T1* given by Equation (21) becomes the optimal cycle time T
if and only if
T1* t M / (1 D ). We substitute Equation (21) into T1* t M / (1 D ) then T *
T1* if and
only if
§ M ·
2 A pI e D M 2 (1 E ) N 2 D ¨
¸
© (1 D ) ¹
2
h cI p d 0
Similarly, T2* given by Equation (22) becomes the optimal cycle time T * if and only if
M d T2* d M / (1 D ). We substitute Equation (22) into M d T2* d M / (1 D ), then T *
T2*
if and only if
2 A pI e D M 2 (1 E ) N 2 M 2 D h cI p (1 D ) 2 d 0
and
2 A pI e D M (1 E ) N
2
2
§ M ·
D¨
¸
© (1 D ) ¹
2
h cI p t 0
Also T3* , given by Equation (23), becomes the optimal cycle time T * if and only if
N d T3* d M .
We substitute Equation (23) into N d T3* d M , then T * T3* if and only if
2 A pI e D M 2 (1 E ) N 2 M 2 D h cI p (1 D )2 t 0
and
2 A N 2 D h cI p (1 D ) 2 pI e E ) d 0
Finally, T4* given by Equation (24) becomes the optimal cycle time T * if and only if
T4* d N .
We substitute Equation (24) into T4* d N , then T * T4* if and only if
2 A N 2 D h cI p (1 D ) 2 pI e E t 0
Furthermore, let
§ M ·
D¨
© (1 D ) ¹¸
2
h cI
'1
2 A pI e D M (1 E ) N
'2
2 A pI e D M 2 (1 E ) N 2 M 2 D h cI p (1 D )2
2
2
(25)
p
(26)
289
Optimal retailer’s ordering policies
'3
2 A N 2 D h cI p (1 D )2 pI e E
(27)
Then we have '1 ! ' 2 ! '3 from Equations (25)–(27). Summarising the above
arguments, the optimal cycle time T * can be obtained as follows.
Theorem 2:
1
If ' 3 t 0, then TRC(T * ) TRC(T4* ) and T *
2
If '3 0 and ' 2 ! 0, then TRC(T * ) TRC(T3* ) and T *
T3* .
3
If ' 2 0 and '1 ! 0, then TRC(T * ) TRC(T2* ) and T *
T2* .
4
If '1 0, then TRC(T * ) TRC(T1* ) and T *
T4* .
T1* .
Case 2 Suppose that M N
Let TRCci (Ti* ) 0 for all i = 5, 6. We can obtain
2 A cI p DM 2 pI e D E M 2
T5*
D h cI p 1 (1 D )2
T6*
2A
D h cI p (1 D )2 pI e E
if 2 A cI p DM 2 pI e DM 2 E ! 0
(28)
(29)
Equation (28) gives the optimal value of T * for the case M d T so that M d T5* . We
substitute Equation (28) into M d T5* .
We yield that
M d T5*
if and only if
2 A M D h cI p (1 D ) pI e E d 0.
2
2
Similarly, Equation (29) gives the optimal value of T * for the case M t T so that
M t T6* . We substitute Equation (29) into M t T6* , and then we find that M t T6* if and
only if 2 A M 2 D h cI p (1 D )2 pI e E t 0.
Furthermore
'4
2 A M 2 D h cI p (1 D )2 pI e E
(30)
From the above arguments, we obtain the following results.
Theorem 3:
1
If ' 4 t 0, then TRC(T * ) TRC(T6* ) and T *
T6* .
2
If ' 4 0, then TRC(T * ) TRC(T5* ) and T *
T5* .
290
5
C.K. Jaggi, K.K. Aggarwal and M. Verma
Special cases
5.1 Huang (2005) model
When N = 0 and therefore E
0 and p t c. Let
TRC7 (T )
cD
A DTh
M2
§T
·
cI p D ¨ D M ¸ pI e D
2
2T
T
©2
¹
TRC8 (T )
cD
§ (1 D ) 2 T (T M )2
A DTh
cI p D ¨¨
T
2
2
2T
©
TRC9 (T )
cD
A DTh
(1 D ) 2 T
§ M (T / 2) ·
cI p D
pDI e ¨
¸
T
T
2
2
©
¹
T7*
T8*
T9*
(31)
·
M2
¸¸ pI e D
2T
¹
2 A pI e DM 2
D h cI p
D h cI p 1 (1 D ) 2
(33)
(34)
2 A cI p DM 2 pI e DM 2
(32)
(35)
(36)
2A
D h cI p (1 D )2 pI e
Then TRCci (Ti* ) 0 for i = 7–9. Equation (5a)–(5d) will be modified as follows:
TRC(T )
M
°TRC7 (T ) if 1 D d T
°
M
°
®TRC8 (T ) if M d T d
1
D
°
°TRC9 (T ) if 0 d T d M
°
¯
(37a)
(37b)
(37c)
Equation (37a)–(37c) will be consistent with Equation (5a)–(5d), in Huang (2005),
respectively. Equations (25) and (26) can be modified as
M2
'1*
2 A pI e DM 2
D h cI p
'*2
2 A pI e DM 2 M 2 D h cI p (1 D ) 2
(1 D ) 2
and
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Optimal retailer’s ordering policies
Thus, Theorem 2 reduces to:
Theorem 4:
1
If '1* ! 0 and '*2 ! 0, then TRC(T * ) TRC(T9* ) and T *
T9* .
2
If '1* ! 0 and '*2 0, then TRC(T * ) TRC(T8* ) and T *
T8* .
3
If '1* 0 and '*2 0, then TRC(T * ) TRC(T7* ) and T *
T7* .
Theorem 4 has been discussed in Theorem 2 of Huang (2005). Hence, Huang (2005) will
be a special case of this paper.
5.2 Huang (2003) model
When M t N , p c, D 1, E 0 (which means supplier as well as retailer offers full
trade credit to his/her customer), let
TRC10 (T )
cD
D M 2 N2
A DTh
D(T M )2
cI p
cI e
T
2
2T
2T
TRC11 (T )
cD
§ T2 N2
·
A DTh
cDI e ¨
(M T ) ¸
¨
¸
T
2
2T
©
¹
(39)
TRC12 (T )
cD
A DTh
cI e D ( M N )
2
T
(40)
(38)
T10*
2 A cD ª M 2 I p I e N 2 I e º
¬
¼
D h cI p
(41)
T11*
2 A cDN 2 I e
D h cI e
(42)
T12*
2A
Dh
(43)
Then TRCci (Ti* ) 0 for i = 10–12. Equation (5a–(5d) will be modified as follows:
TRC(T )
TRC (T ) if M d T
10
°°
®TRC11(T ) if N d T d M
°
°¯TRC12 (T ) if T t N
(44a)
(44b)
(44c)
Equation (44a)–(44c) will be consistent with Equation (5a)–(5d), in Huang (2003),
respectively. Equations (26) and (27) can be modified as ' 2 2 A cI e DN 2
M D h cI e and ' 3
2
2 A N Dh, and hence Theorem 2 reduces to:
2
292
C.K. Jaggi, K.K. Aggarwal and M. Verma
Theorem 5:
1
If ' 2 ! 0 and '3 t 0 then TRC(T * ) TRC(T12* ) and T *
T12* .
2
If ' 2 ! 0 and '3 0, then TRC(T * ) TRC(T11* ) and T *
T11* .
3
If ' 2 d 0 and '3 0, then TRC(T * ) TRC(T10* ) and T *
T10* .
Theorem 5 has been discussed in Theorem 1 of Huang (2003). Hence, Huang (2003) will
be a special case of this paper.
5.3 Goyal (1985) model
When N = 0, which means the supplier would offer the retailer a delay period but the
retailer would not offer the delay period to his/her customer. Therefore, when p c, and
D 1 and E 0. Let
TRC13 (T )
cD
A DTh
D(T M )2
DM 2
cI p
cI e
2
2T
2T
T
(45)
TRC14 (T )
cD
A DTh
ªT
º
cDI e « ( M T ) »
2
2
T
¬
¼
(46)
2 A cDM 2 I p I e
T13*
D h cI p
(47)
2A
D h cI e
T14*
(48)
Then TRCci (Ti* ) 0 for i = 13, 14. Equation (5a)–(5d) will be modified as follows:
TRC(T )
TRC13 (T ) if M d T
®
¯TRC14 (T ) if M t T
(49a)
(49b)
Equation (49a) and (49b) will be consistent with Equation (5a)–(5d), in Goyal (1985),
respectively. Equation (26) can be modified as ' 2 A M 2 D (h cI e ), and Theorem 2
reduces to:
Theorem 6:
1
If ' ! 0 then TRC(T * ) TRC(T11* ) and T *
2
If ' 0, then TRC(T * ) TRC(T10* ) and T *
3
If '
0, then T *
T13*
T14*
T14* .
T13* .
M.
Theorem 6 has been discussed in Theorem 1 of Chung (1998). Hence, Goyal (1985) will
be a special case of this paper.
293
Optimal retailer’s ordering policies
6
Computational analysis
The purposes of this numerical analysis are:
1
to obtain optimal solutions for the two cases
2
to compare the results of present analysis with the previous research work
3
to use sensitivity analysis to highlight the influence of model parameters.
6.1 Numerical analysis
Case 1
M tN
Example 1: To gain more insight into the above theory, we borrowed Huang and Hsu
(2008) example, which is stated as:
A = 80, D = 2000, c = 10, p = 30, h = 7, I p = 0.15, I e = 0.13, M = 0.1, N = 0.08,
E
0.2 and D = 0.09 in appropriate units.
The optimal value of T * and Q* for the present study are T * 0.09454 year and Q* 189
units.
Further, the comparative analysis of proposed model with the previous research work
has been summarised in Table 1.
It is clearly apparent from Table 1 that by implementing partial trade credit policy,
the retailer is able to increase his ordering quantity, which implies that he is able to attract
more customers, which otherwise was not possible, as evident from the Table 1. Hence,
offering partial trade credit at two levels of the supply chain is beneficial for the retailer.
Case 2
M N
Example 2: Let A = 80, D = 5000, c = 10, h =10, I p
and E
0.2, D
0.1, I e
0.2, M = 0.05, N=0.06
0.1 in appropriate units.
The optimal value of T * and Q* for the present model are T *
Q
*
0.057023 year and
285 units.
The sensitivity analysis on different parameters has been presented in Section 6.2.
6.2 Sensitivity analysis
In any decision-making situation, the change in the values of parameters may happen due
to uncertainties. To examine the implications of these changes, the sensitivity analysis
will be of great help in decision-making.
Firstly, we investigate the changing effects of parameters D , p and N for fixed E ,
when M t N in Table 2. Then Table 3 shows the effect of changes in E , p and N for
fixed D for the same case.
From Tables 2 and 3, we have the following inferences:
x
For fixed p and N, an increases in D implies that the retailer’s initial payment to the
supplier decreases, results an increase in the retailer’s order quantity.
294
C.K. Jaggi, K.K. Aggarwal and M. Verma
x
For fixed N and D , when p increases, the optimal cycle length for the retailer
decreases, which in a way increases his inventory turnover. This implies that the
retailer will be ordering more frequently, which eventually helps him to generate
more revenue on the initial payment received from the customer.
x
For fixed p and D , as N increases, i.e. the credit period offered by the retailer to his
customer increases, retailer’s cycle length increases so as his order quantity. This
implies that the retailer is able to attract more and more customers, and he will be
able to generate more revenue on the initial payment received from the customers.
Lastly, for the case M N , Table 4 examines the changes in the parameter D , p and N
for fixed E . It is revealed from Table 4 that for fixed N and ( 1 D ), as p increases,
there is a decrease in the annual total cost.
Table 1
Comparative analysis
T*
Q*
Present study
0.09454
189
Huang (2005)
0.0812
162
0.08648
173
0.08648
173
Research article
N 0, E 0
Huang (2003)
p
c, D
1, E
0
Goyal (1985)
p
c, N
Table 2
1D
0.9
0, D
1, E
0
Optimal solutions when M t N , E fixed
N
p
0.02
10
30
50
10
30
50
10
30
50
10
30
50
10
30
50
0.05
0.08
0.5
0.02
0.05
T*
0.091
0.081
0.074
0.093
0.085
0.079
0.095
0.090
0.087
0.096
0.084
0.076
0.097
0.088
0.081
Q*
TRC (T * ) (in $)
183
164
149
186
170
159
191
182
175
193
170
154
195
176
164
21,489.47
21,204.26
20,898.01
21,513.06
21,282.71
21,039.65
21,556.12
21,421
21,282.18
21,410.46
21,134.23
20,834.35
21,432.99
21,209.92
20,971.89
295
Optimal retailer’s ordering policies
Table 2
1D
0.1
Optimal solutions when M t N , E fixed (continued)
N
p
T*
Q*
0.08
10
0.099
200
21,474.06
30
0.094
188
21,343.3
50
0.090
181
21,207.4
0.02
0.05
0.08
Table 3
10
0.098
197
21,375.43
30
0.086
173
21,103.42
50
0.077
156
20,806.48
10
0.099
199
21,397.49
30
0.089
179
21,177.89
50
0.082
166
20,942.23
10
0.10
204
21,437.71
30
0.09
191
21,309.2
50
0.09
183
21,174.65
Optimal solutions when M t N , D fixed
E
N
0.2
0.02
0.05
0.08
0.4
TRC (T * ) (in $)
0.02
0.05
0.08
p
T*
Q*
TRC (T * ) (in $)
10
0.098
197
21,375.43
30
0.086
173
21,103.42
50
0.077
156
20,806.48
10
0.099
199
21,397.49
30
0.089
179
21,177.89
50
0.082
166
20,942.23
10
0.102
204
21,437.71
30
0.095
191
21,309.2
50
0.091
183
21,174.65
10
0.098
197
21,374.37
30
0.086
172
21,099.8
50
0.077
155
20,799.79
10
0.099
199
21,390.96
30
0.088
177
21,156.03
50
0.081
163
20,902.7
10
0.101
202
21,421.32
30
0.093
187
21,256.3
50
0.088
176
21,082.04
296
C.K. Jaggi, K.K. Aggarwal and M. Verma
Table 3
Optimal solutions when M t N , D fixed (continued)
E
N
0.6
0.02
p
0.05
0.08
Table 4
T*
Q*
TRC (T * ) (in $)
10
0.098
196
21,373.32
30
0.085
172
21,096.18
50
0.077
155
20,793.09
10
0.098
198
21,384.4
30
0.087
175
21,133.92
50
0.080
160
20,862.45
10
0.100
200
21,404.78
30
0.099
182
21,202.1
50
0.084
169
20,985.69
Optimal solutions when M N , E fixed
1D
p
T*
Q*
TRC (T * ) (in $)
0.9
10
0.054988
275
52,885.93
30
0.057023
285.1
52,772.93
50
0.04998
250
52,701.44
0.5
0.1
7
10
0.056389
282
52,807.3
30
0.05859
293
52,691.03
50
0.061072
305.3
52,569.56
10
0.057023
285.1
52,772.93
30
0.059303
296.5
52,655.17
50
0.06188
309.3
52,531.99
Conclusion
This paper extends the assumptions of the two-level trade credit policy in the previously
published results to reflect the realistic situations by incorporating partial trade credit.
It is assumed that the supplier as well as the retailer adopts partial trade credit financing
to stimulate their demand to get optimal inventory policy. The theorems help the retailer
to yield the optimal ordering policy efficiently. Further, we have deduced the models
presented by Huang (2005, 2003) and Goyal (1985). It has been shown that as the
retailer’s initial payment to the supplier decreases, his ordering quantity increases.
Further, for the case M t N , an increase in N results in an increase in his order
quantity, which implies that he is able to attract more customers. Moreover, as the selling
price increases, retailer’s order cycle decreases. This indicates that he should order more
frequently and take the advantage of earning more revenue from the initial payment
received from the customers. The findings have been validated with the help of the
examples. Further, comprehensive sensitive analysis along with a case study has also
been presented.
Optimal retailer’s ordering policies
297
A future study can be incorporated in the present model into more realistic
assumptions such as probabilistic demand, allowable shortages, deteriorating items or
finite replenishment rate.
Acknowledgements
The authors really appreciate the constructive comments by the anonymous referees,
which greatly helped in improving this paper further. The first author would like to
acknowledge the support of Research Grant No. Dean (R)/R&D/2009/487, provided by
University of Delhi, Delhi, India, for conducting this research. The third author would
like to thank University Grant Commission (UGC) for providing the fellowship to
accomplish the research.
References
Abad, P.L. and Jaggi, C.K. (2003) ‘A joint approach for setting unit price and the length of the
credit period for seller when end demand is price sensitive’, Int. J. Production Economics,
Vol. 83, pp.115–122.
Aggarwal, S.P. and Jaggi, C.K. (1995) ‘Ordering policies of deteriorating items with permissible
delay in payments’, Journal of Operational Research Society, Vol. 46, pp.658–662.
Chang, H.J. and Dye, C.Y. (2001) ‘An inventory model for deteriorating items with partial
backlogging and permissible delay in payments’, Int. J. System Science, Vol. 32, pp.345–352.
Chung, K.J. (1998) ‘A theorem on the determination of economic order quantity under conditions
of permissible delay in payments’, Computers and Operations Research, Vol. 25, pp.49–52.
Chung, K.J., Goyal, S.K. and Huang, Y.F. (2005) ‘The optimal inventory policies under
permissible delay in payments depending on the ordering quantity’, Int. J. Production
Economics, Vol. 95, pp.203–213.
Chung, K.J. and Liao, J.J. (2006) ‘The optimal ordering policy in a DCF analysis for deteriorating
items when trade credit depends on the order quantity’, Int. J. Production Economics,
Vol. 100, pp.116–130.
Goyal, S.K. (1985) ‘EOQ under conditions of permissible delay in payments’, Journal of
Operational Research Society, Vol. 36, pp.335–338.
Huang, Y.F. (2003) ‘Optimal retailer ordering policies in the EOQ model under trade credit
financing’, Journal of Operational Research Society, Vol. 54, pp.1011–1015.
Huang, Y.F. (2005) ‘Retailer’s inventory policy under supplier’s partial trade credit policy’,
Journal of the Operations Research Society of Japan, Vol. 48, pp.173–182.
Huang, Y.F. (2007) ‘Optimal retailer’s replenishment decisions in the EPQ model under two levels
of trade credit policy’, European Journal of Operational Research, Vol. 176, pp.1577–1591.
Huang, Y.F. and Hsu, K.H. (2008) ‘An EOQ model under retailer partial trade credit policy in
supply chain’, Int. J. Production Economics, Vol. 112, pp.655–664.
Hwang, H. and Shinn, S.W. (1997) ‘Retailers pricing and lot sizing policies for exponentially
deteriorating products under the condition of permissible delay in payments’, Computers and
Operations Research, Vol. 24, pp.539–547.
Jamal, A.M.M., Sarkar, B.R. and Wang, S. (1997) ‘An ordering policy for deteriorating items with
allowable shortages and permissible delay in payment’, Journal of the Operational Research
Society, Vol. 48, pp.826–833.
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Jamal, A.M.M., Sarkar, B.R. and Wang, S. (2000) ‘Optimal payment time for a retailer under
permitted delay in payment by the wholesaler’, Int. J. Production Economics, Vol. 66,
pp.59–66.
Jaggi, C.K., Goyal, S.K. and Goel, S.K. (2008) ‘Retailer’s optimal replenishment decisions with
credit-linked demand under permissible delay in payments’, European Journal of Operational
Research, Vol. 190, pp.130–135.
Thangam, A. and Uthayakumar, R. (2008) ‘Analysis of partial trade credit financing in a supply
chain by EPQ based models’, Advanced Modeling and Optimization, Vol. 10, pp.177–198.
Optimal retailer’s ordering policies
299
Appendix
Case Illustration
A retailer purchases medicine from a pharmaceutical company at c = $100 per unit
and sells it at p = $120 per unit. The company offers a partial credit (i.e. D 0.09 ) for the
time period M = 0.1 year. However, if the payment is not made within the specified
period then 15% interest (i.e. I p 0.15 ) is charged on the outstanding amount. To avoid
default risks, the retailer also offers a partial trade credit (i.e. E 0.02 and N = 0.05 year)
to the credit risk customers. The retailer earns interest at the rate I e 10% on the revenue
generated. Also, the annual demand for the retailer is D = 2,000 units with holding cost
h = $5 per unit per year and the ordering cost A = $250 per order.
Analysis
Using the above proposed model, it is apparent that if both the pharmaceutical company
as well as the retailer offers partial trade credit to its subsequent downstream member,
then the optimal cycle length and order quantity for the retailer is T * 0.1418 year
and Q* 283, respectively. However, when the retailer does not offer any credit period
(i.e. E = 0 and N = 0) to his/her customer, then his cycle length and order quantity
becomes T * 0.1409 year and Q* 281 units, respectively, which implies that the
retailer is able to attract more customers by incorporating partial trade credit, which in
turn increases his order quantity. Thus, partial trade credit is beneficial from the retailer’s
viewpoint.
Optimal retailer’s ordering policies under two-stage
partial trade credit financing in a supply chain
Chandra K. Jaggi*, K.K. Aggarwal and
Mona Verma
Department of Operational Research,
Faculty of Mathematical Sciences,
New Academic Block,
University of Delhi,
Delhi 110007, India
Fax: +91 11 27666672
E-mail: [email protected]
E-mail: [email protected]
*Corresponding author
Abstract: In this paper, an attempt has been made to investigate partial trade
credit financing for two levels of supply chain, i.e. the retailer as well as the
customer must make a partial payment initially in order to make them eligible
for availing the delay period for the rest of their purchases. We have developed
the retailer’s inventory system as a cost minimisation problem to determine the
retailer’s optimal ordering policies. Further, we have deduced the models
presented by Huang (2005, 2003) and Goyal (1985). It has been established
numerically that offering partial trade credit in two-stage supply chain is
beneficial as per retailer’s perspective. Comprehensive sensitive analysis along
with a case study has also been presented.
Keywords: partial trade credit; EOQ; economic order quantity; inventory;
supply chain.
Reference to this paper should be made as follows: Jaggi, C.K.,
Aggarwal, K.K. and Verma, M. (2012) ‘Optimal retailer’s ordering policies
under two-stage partial trade credit financing in a supply chain’, Int. J.
Industrial and Systems Engineering, Vol. 10, No, 3, pp.277–299.
Biographical notes: Chandra K. Jaggi is an Associate Professor in the
Department of Operational Research, University of Delhi, India. He earned his
PhD and MPhil in Inventory Management and Masters in Operational Research
from Department of Operational Research, University of Delhi. His area of
teaching includes inventory and financial management. His area of research is
production/inventory and supply chain management. He has publications in Int.
J. Production Economics, Journal of Operational Research Society, European
Journal of Operational Research, Int. J. Systems Sciences, Int. J. Procurement
Management, TOP, Int. J. Applied Decision Sciences, OPSEARCH,
Investigacion Operacional Journal, Int. J. Operational Research, Advanced
Modeling and Optimization, Canadian Journal of Pure and Applied Sciences,
Journal of Information and Optimization Sciences, Int. Journal of
Mathematical Sciences, Indian Journal of Mathematics and Mathematical
Sciences, Indian Journal of Management and Systems, etc.
Copyright © 2012 Inderscience Enterprises Ltd.
277
278
C.K. Jaggi, K.K. Aggarwal and M. Verma
K.K. Aggarwal is an Assistant Professor in the Department of Operational
Research, University of Delhi, India. He obtained his PhD in Inventory
Management and Masters in Operational Research from Department of
Operational Research, University of Delhi. Prior to his appointment as
Assistant Professor, he served as Scientist in National Informatics Centre
(India). His research interests and teaching include inventory modelling,
financial engineering and network analysis. He has published more than 15
research papers in Int. J. Production Economics, Indian Journal of
Mathematics and Mathematical Sciences and Investigacion Operacional
Journal.
Mona Verma is a Research Scholar in the Department of Operational Research,
Faculty of Mathematical Sciences, University of Delhi, India. She has
completed her MPhil in Inventory Management and Masters in Operational
Research from the Department of Operational Research, University of Delhi.
Currently, she is pursuing her PhD in Operational Research. Her research
interest lies in inventory management and supply chain management.
1
Introduction
In today’s business transactions, it is generally observed that the retailers are allowed a
fixed time period before they settle their account with the supplier, which is simply
known as trade credit period. During this period, the retailer can sell the goods and earn
interest on revenue generated. After this period, if the account is not settled, then the
supplier charges interest on the remaining inventory as per the terms and conditions
agreed upon. This is termed as one-level trade credit financing, which has been explored
by many researchers at length. Initially, Goyal (1985) had analysed the effects of trade
credit on the optimal inventory policy. Chung (1998) developed an alternative approach
to determine the EOQ under the condition of permissible delay in payments. Further,
Aggarwal and Jaggi (1995) considered the inventory model with an exponential
deterioration rate under the condition of permissible delay in payments. Jamal et al.
(1997) and Chang and Dye (2001) further generalised the model with shortages. Further,
Hwang and Shinn (1997), Jamal et al. (2000), Chung et al. (2005), Chung and Liao
(2006) also presented their work under varying assumptions. Abad and Jaggi (2003) first
offered a supplier–retailer integrated model following a lot-for-lot shipment policy under
permissible delay in payment. In their model, the retailer takes advantage of the trade
credit offered by the supplier but requires his/her customers to pay for the items
immediately. In all the above papers, researchers have assumed that the retailer does not
offer credit period to his/her customer. Huang (2003) presented an inventory model
assuming that the retailer also offers a credit period to his/her customer which is shorter
than the credit period offered by the supplier to him, in order to stimulate his own
demand. This type of trade credit is termed as two-level trade credit policy in a supply
chain modelling. Later on, Huang (2007) investigated the optimal retailer’s replenishment
decisions with two levels of trade credit policy in the EPQ framework. Ho et al. (2007)
also developed an integrated supplier retailer inventory model with demand rate increases
exponentially with the customer’s credit period. Further, Jaggi et al. (2008) incorporated
the concept of credit-linked demand and developed an inventory model under two levels
of trade credit policy to determine the optimal credit as well as replenishment policy
Optimal retailer’s ordering policies
279
jointly, for the retailer. Another realistic phenomenon of the supply chain modelling is
partial trade credit, i.e. the retailer/customer need to make some partial amount for their
purchases in order to make themselves eligible for availing the credit period for the rest
of the amount .Huang (2005) developed optimal retailer’s policy for one level where the
supplier offers partial trade credit to the retailer. Adding to this, Huang and Hsu (2008)
expanded the model where retailer gets full trade credit but offers partial trade credit to
his/her customer. Thangam and Uthayakumar (2008) developed the partial trade credit
financing in an EPQ model under the same environment.
Unfortunately, none of the researchers have incorporated the partial trade credit for
supplier as well as retailer. In this paper, an attempt has been made to investigate partial
trade credit financing for a two level of supply chain, i.e. the retailer as well as the
customer must make a partial payment initially in order to make them eligible for
availing the delay period for the rest of their purchases. We have developed the retailer’s
inventory system as a cost minimisation problem to determine the retailer’s optimal
ordering policies. Further, we have deduced the models presented by Huang (2003, 2005)
and Goyal (1985). It has been established numerically that offering partial trade credit in
two-stage supply chain is beneficial from the retailer’s perspective.
2
Assumptions and notations
We formulate a two-echelon supply chain model, for the general case, where the supplier
as well as the retailer offers partial trade credit to their downstream mentor. The case of
full credit then is just the special case of the general model. The nomenclature is as
follows:
D demand rate per year
A
ordering cost per order
c
unit purchase price for retailer
p
unit selling price for retailer, c d p
h
unit stockholding cost per year excluding interest charges
I e interest earned per $ per year for retailer
I p interest charged per $ in stocks per year for retailer
M retailer’s credit period offered by the supplier for settling the accounts
N customer’s credit period offered by the retailer for settling the account
D
the percentage of permissible delay in payments for retailer, 0 d D d 1
E
the percentage of permissible delay in payments for customer, 0 d E d 1
T
cycle time in years
TRC(T ) annual total relevant cost, which is a function of T .
280
C.K. Jaggi, K.K. Aggarwal and M. Verma
The following assumptions are made in the model:
1
Demand rate is known and constant.
2
Shortages are not allowed.
3
Time horizon is infinite.
4
The supplier offers a partial trade credit period M to the retailer. As the order is
received, the retailer has to make a partial payment of (1 D )cDT to the supplier.
During this period, the retailer earns interest
I e (d, t, I p ) on the revenue
generated from sales at the selling price p ! c, whereas if the account is not settled
at M , then the supplier charges interest I p on the remaining stock at the purchase
price c.
5
The retailer also offers a partial trade credit period N to his/her customer, i.e. his/her
customer must make a partial payment E pDT to the retailer, and they must pay off
the rest of the amount at the end of the trade credit period N (independent of M ),
which is offered to him by the retailer. Now the retailer earns interest I e on the
partial payment, which he receives from his/her customer.
3
The retailer’s problem
The retailer wants to stimulate his sales by offering partial trade credit E to his/her
customer, while making himself eligible for the same by giving an initial payment on
(1 D ) units to the supplier .The objective is to minimise annual total cost, which
comprises of the following elements:
Annual purchasing cost = cD
Annual ordering cost = A/T
Annual stockholding cost (excluding interest charges) = DTh/2
The annual total relevant cost for the retailer can be expressed as shown below:
TRC(T) = Annual purchase cost + Annual ordering cost + Annual stockholding
cost + Annual interest payable – Annual interest charged
After the credit period M , the retailer has to make payment on the items in stock at
the interest I p and he will earn interest at the rate I e if the customer does not pay for the
items at the specified credit period N offered by him. The two situations that may arise
are as follows:
1
M tN
2
M N
Case 1 M t N : According to assumption 4 and 5, there are four subcases to occur for
interest payable per year and interest earned per year for the retailer.
1
N dM d
M
d T (Figure 1)
1D
281
Optimal retailer’s ordering policies
In this case, since the retailer makes a partial payment of (1 D ) units to the supplier, the
interest paid, therefore, will be on the balance amount of purchases made by him.
Annual interest payable
Annual interest earned
ª DT 2 / 2 D DTM º
»
cI p «
«
»
T
¬
¼
ª E DN 2 / 2 (( DN DM )( M N ) / 2)) º
»
pI e «
«
»
T
¬
¼
Therefore, the annual total relevant cost will be
TRC1 (T )
2
M dT d
cD
ª
M 2 N2
E N2
A DTh
ªT
º
cI p D « D M » pI e D «
« 2T
2
2T
T
¬2
¼
¬
º»
»
¼
(1)
M
(Figure 2)
1D
Annual interest payable
ª (1 D )2 DT 2 / 2 D(T M )2 / 2 º
»
cI p «
«
»
T
¬
¼
ª (1 D ) 2 T 2 (T M )2 º
cDI p «
»
2T
¬
¼
Annual interest earned
ª E DN 2 / 2 (( DN DM )( M N )) / 2 º
»
pI e «
«
»
T
¬
¼
Thus, the annual total relevant cost is given by
TRC 2 (T )
cD
ª (1 D ) 2 T (T M ) 2 º
A DTh
cI p D «
»
T
2
2
2T
¬
¼
ª
M 2 N2
E N2
pI e D «
« 2T
2T
¬
3
º»
(2)
»
¼
N d T d M (Figure 3)
Annual interest payable
ª (1 D ) 2 DT 2 / 2 º
»
cI p «
«
»
T
¬
¼
Annual interest earned
ª E DN 2 / 2 (( DN DT )(T N )) / 2 DT ( M T ) º
»
pI e «
«
»
T
¬
¼
282
C.K. Jaggi, K.K. Aggarwal and M. Verma
Hence, the annual total relevant cost will be
TRC3 (T )
A DTh
(1 D ) 2 T
cI p D
T
2
2
2
2
ª
º
EN 2 T N
pDI e «
( M T )»
« 2T
»
2T
¬
¼
cD
4
T d N (Figure 4)
º»
Annual interest payable
ª (1 D ) 2 DT 2 / 2
cI p «
«
T
¬
Annual interest earned
ª E DT 2 / 2 ( E DT ( N T )) ( DT ( M N )) º
»
pI e «
«
»
T
¬
¼
Figure 1
N d M / (1 D ) d T
Figure 2
M d T d M / (1 D )
»
¼
(3)
Optimal retailer’s ordering policies
Figure 3
N dT d M
Figure 4
TdN
283
Consequently, the annual total relevant cost is given as
TRC4 (T )
cD
A DTh
(1 D ) 2 T
ET
ª
º
cI p D
pI e D « M
(1 E ) N »
T
2
2
2
¬
¼
(4)
From the above arguments, the annual total relevant cost for the retailer can be
expressed as
TRC(T )
°TRC1 (T )
°
°TRC (T )
2
®
°
°TRC3 (T )
°TRC (T )
¯
4
if
M
dT
1 D
M
1 D
if N d T d M
if M d T d
if T t N
(5a)
(5b)
(5c)
(5d)
284
C.K. Jaggi, K.K. Aggarwal and M. Verma
Since TRC1 ( M / (1 D )) TRC2 ( M / (1 D )), TRC2 ( M ) TRC3 ( M ) and TRC3 ( N )
TRC4 ( N ), TRC(T ) is continuous and well defined. All TRCi (T ) i 1 4 and TRC(T )
are defined on T > 0. To minimise the annual total cost, we take the first order and second
order derivatives of TRCi (T ) i 1 4 with respect to T . Therefore, Equations (1)–(4)
yield
ª 2 A pI e D M 2 (1 E ) N 2
«
«
2T 2
¬
TRC1cc(T )
ª 2 A pI e D M 2 (1 E ) N 2
«
«
T3
¬
TRC 2c (T )
ª 2 A cI p DM 2 pI e D M 2 (1 E ) N 2
«
«
2T 2
¬
»
¼
«
¬
2
P
º
»
¼
(6)
º»
(7)
»
¼
ª h cI p 1 (1 D ) 2
D«
«
2
¬
ȼ
º»
»
¼
(8)
»
¼
º»
TRCcc2 (T )
ª 2 A cI p DM 2 pI e D M 2 (1 E ) N 2
«
«
T3
¬
TRC3c (T )
ª h cI p (1 D ) 2 pI e º
ª 2 A pI e DN 2 (1 E ) º
«
»
«
D
»
2
2T 2
«¬
»¼
¬«
¼»
(10)
TRC3cc (T )
ª 2 A pI e DN 2 (1 E ) º
«
»!0
T3
«¬
»¼
(11)
TRCc4 (T )
ª h cI p (1 D ) 2 pI e E º
ª Aº
»
« 2 » D«
2
¬T ¼
«¬
»¼
(12)
TRCcc4 (T )
Let [1
»º D ª h cI
TRC1c (T )
2A
T3
(9)
»
¼
!0
2 A pI e D( M 2 (1 E ) N 2 ), [2
(13)
2 A cI p DM 2 pI e D ( M 2 (1 E ) N 2 ).
Then we can find that [1 d [2 . Equations (11) and (13) imply that TRC3 (T ) and
TRC4 (T ) is convex on T > 0. However, TRC1 (T ) is convex on T > 0 if [1 ! 0
and TRC2 (T ) is convex on T > 0 if [2 ! 0. Furthermore, we have
§ M ·
TRC1c ¨
¸
© 1 D ¹
§ M ·
TRCc2 ¨
¸ , TRCc2 ( M )
©1D ¹
TRCc3 ( M ) and TRCc3 ( N )
TRCc4 ( N )
285
Optimal retailer’s ordering policies
Therefore, Equation (5a)–(5d) imply that TRC(T ) is convex on T > 0 if [1 ! 0. Then we
can obtain the following results.
Theorem 1:
1
If [2 d 0, then TRC (T ) is convex on (0, M ] and concave in [ M , f).
2
If [ 1 t 0 and [2 ! 0, then TRC (T ) is convex on (0,
M
] and concave in
1D
M
[
, f ).
1 D
3
If [1 d 0, then TRC (T ) is convex on (0, f).
Case 2
M N
Annual ordering cost = A/T
Annual stockholding cost (excluding interest charges) = DTh/2
According to assumption 5 and assumption, there are two cases to occur in interest
payable per year and interest earned per year, i.e.
1
M dT
2
M tT
1
M d T (Figure 5)
Annual interest payable
Annual interest earned
2
ª (1 D ) 2 T (T M )2 º
cI p D «
»
2
2T
¬
¼
pI e
E DM 2
2T
M t T (Figure 6)
Annual interest payable
Annual interest earned
ª (1 D )2 T º
cI p D «
»
2
¬
¼
ª E DT 2
º
pI e «
E DT ( M T ) »
¬ 2
¼
Tº
ª
pI e E D « M »
2¼
¬
From the above arguments, the annual total relevant cost for the retailer can be
expressed as
286
C.K. Jaggi, K.K. Aggarwal and M. Verma
Figure 5
Interest earned when M d T
Figure 6
Interest earned when M t T
TRC(T) = Annual purchase cost + Annual ordering
cost + Annual interest payable – Annual interest charged
where
TRC(T )
cost + annual
stockholding
TRC5 (T ) if M d T
®
¯TRC6 (T ) if M t T
(14a)
(14b)
where
TRC5 (T )
cD
ª (1 D )2 T (T M ) 2 º pI e D E M 2
A DTh
cI p D «
»
2
2
2T
2T
T
¬
¼
(15)
287
Optimal retailer’s ordering policies
and
cD
TRC6 (T )
ª (1 D )2 T º
A DTh
Tº
ª
cI p D «
» pI e D E « M »
T
2
2
2¼
¬
¬
¼
(16)
Since TRC5 ( M ) TRC6 ( M ), TRC(T ) is continuous and well defined. All TRCi (T ),
i 5, 6 and TRC(T ) are defined on T > 0. Equations (15) and (16) yield
TRC5c (T )
ª h cI p 1 (1 D ) 2
ª 2 A cI p DM 2 pI e D E M 2 º
» D«
«
«
2
2T 2
«¬
»¼
¬
TRCcc5 (T )
ª 2 A cI p DM 2 pI e D E M 2 º
«
»
T3
«¬
»¼
TRCc6 (T )
2
ª A º D h cI p (1 D ) pI e E
« 2 »
2
¬T ¼
TRC6cc (T )
ª2Aº
«T3 » ! 0
¬ ¼
º»
»
¼
(17)
(18)
and
(19)
(20)
Equation (20) implies that TRC6 (T ) is convex on T > 0 and Equation (18) implies that
TRC5 (T ) will be convex on T > 0 when 2 A cI p DM 2 pI e DM 2 E ! 0. Furthermore, we
have TRC5c ( M ) TRC6c ( M ). Therefore, Equation (14a) and (14b) implies that TRC(T )
is convex on T > 0 when 2 A cI p DM 2 pI e DM 2 E ! 0.
4
Determination of the optimal cycle time T *
Suppose that M t N
Let TRCci (Ti* ) 0 for all i = 1–4. We can obtain
Case 1
T1*
T2*
T3*
2 A pI e D M 2 (1 E ) N 2
D h cI p
,
if [1 ! 0, [2 ! 0
(21)
,
(22)
2 A cI p DM 2 pI e D M 2 (1 E ) N 2
D h cI p 1 (1 D ) 2
2 A pI e D(1 E ) N 2
D h cI p (1 D ) 2 pI e
if [ 2 ! 0
(23)
288
C.K. Jaggi, K.K. Aggarwal and M. Verma
T4*
2A
D h cI p (1 D )2 pI e E
(24)
T1* given by Equation (21) becomes the optimal cycle time T
if and only if
T1* t M / (1 D ). We substitute Equation (21) into T1* t M / (1 D ) then T *
T1* if and
only if
§ M ·
2 A pI e D M 2 (1 E ) N 2 D ¨
¸
© (1 D ) ¹
2
h cI p d 0
Similarly, T2* given by Equation (22) becomes the optimal cycle time T * if and only if
M d T2* d M / (1 D ). We substitute Equation (22) into M d T2* d M / (1 D ), then T *
T2*
if and only if
2 A pI e D M 2 (1 E ) N 2 M 2 D h cI p (1 D ) 2 d 0
and
2 A pI e D M (1 E ) N
2
2
§ M ·
D¨
¸
© (1 D ) ¹
2
h cI p t 0
Also T3* , given by Equation (23), becomes the optimal cycle time T * if and only if
N d T3* d M .
We substitute Equation (23) into N d T3* d M , then T * T3* if and only if
2 A pI e D M 2 (1 E ) N 2 M 2 D h cI p (1 D )2 t 0
and
2 A N 2 D h cI p (1 D ) 2 pI e E ) d 0
Finally, T4* given by Equation (24) becomes the optimal cycle time T * if and only if
T4* d N .
We substitute Equation (24) into T4* d N , then T * T4* if and only if
2 A N 2 D h cI p (1 D ) 2 pI e E t 0
Furthermore, let
§ M ·
D¨
© (1 D ) ¹¸
2
h cI
'1
2 A pI e D M (1 E ) N
'2
2 A pI e D M 2 (1 E ) N 2 M 2 D h cI p (1 D )2
2
2
(25)
p
(26)
289
Optimal retailer’s ordering policies
'3
2 A N 2 D h cI p (1 D )2 pI e E
(27)
Then we have '1 ! ' 2 ! '3 from Equations (25)–(27). Summarising the above
arguments, the optimal cycle time T * can be obtained as follows.
Theorem 2:
1
If ' 3 t 0, then TRC(T * ) TRC(T4* ) and T *
2
If '3 0 and ' 2 ! 0, then TRC(T * ) TRC(T3* ) and T *
T3* .
3
If ' 2 0 and '1 ! 0, then TRC(T * ) TRC(T2* ) and T *
T2* .
4
If '1 0, then TRC(T * ) TRC(T1* ) and T *
T4* .
T1* .
Case 2 Suppose that M N
Let TRCci (Ti* ) 0 for all i = 5, 6. We can obtain
2 A cI p DM 2 pI e D E M 2
T5*
D h cI p 1 (1 D )2
T6*
2A
D h cI p (1 D )2 pI e E
if 2 A cI p DM 2 pI e DM 2 E ! 0
(28)
(29)
Equation (28) gives the optimal value of T * for the case M d T so that M d T5* . We
substitute Equation (28) into M d T5* .
We yield that
M d T5*
if and only if
2 A M D h cI p (1 D ) pI e E d 0.
2
2
Similarly, Equation (29) gives the optimal value of T * for the case M t T so that
M t T6* . We substitute Equation (29) into M t T6* , and then we find that M t T6* if and
only if 2 A M 2 D h cI p (1 D )2 pI e E t 0.
Furthermore
'4
2 A M 2 D h cI p (1 D )2 pI e E
(30)
From the above arguments, we obtain the following results.
Theorem 3:
1
If ' 4 t 0, then TRC(T * ) TRC(T6* ) and T *
T6* .
2
If ' 4 0, then TRC(T * ) TRC(T5* ) and T *
T5* .
290
5
C.K. Jaggi, K.K. Aggarwal and M. Verma
Special cases
5.1 Huang (2005) model
When N = 0 and therefore E
0 and p t c. Let
TRC7 (T )
cD
A DTh
M2
§T
·
cI p D ¨ D M ¸ pI e D
2
2T
T
©2
¹
TRC8 (T )
cD
§ (1 D ) 2 T (T M )2
A DTh
cI p D ¨¨
T
2
2
2T
©
TRC9 (T )
cD
A DTh
(1 D ) 2 T
§ M (T / 2) ·
cI p D
pDI e ¨
¸
T
T
2
2
©
¹
T7*
T8*
T9*
(31)
·
M2
¸¸ pI e D
2T
¹
2 A pI e DM 2
D h cI p
D h cI p 1 (1 D ) 2
(33)
(34)
2 A cI p DM 2 pI e DM 2
(32)
(35)
(36)
2A
D h cI p (1 D )2 pI e
Then TRCci (Ti* ) 0 for i = 7–9. Equation (5a)–(5d) will be modified as follows:
TRC(T )
M
°TRC7 (T ) if 1 D d T
°
M
°
®TRC8 (T ) if M d T d
1
D
°
°TRC9 (T ) if 0 d T d M
°
¯
(37a)
(37b)
(37c)
Equation (37a)–(37c) will be consistent with Equation (5a)–(5d), in Huang (2005),
respectively. Equations (25) and (26) can be modified as
M2
'1*
2 A pI e DM 2
D h cI p
'*2
2 A pI e DM 2 M 2 D h cI p (1 D ) 2
(1 D ) 2
and
291
Optimal retailer’s ordering policies
Thus, Theorem 2 reduces to:
Theorem 4:
1
If '1* ! 0 and '*2 ! 0, then TRC(T * ) TRC(T9* ) and T *
T9* .
2
If '1* ! 0 and '*2 0, then TRC(T * ) TRC(T8* ) and T *
T8* .
3
If '1* 0 and '*2 0, then TRC(T * ) TRC(T7* ) and T *
T7* .
Theorem 4 has been discussed in Theorem 2 of Huang (2005). Hence, Huang (2005) will
be a special case of this paper.
5.2 Huang (2003) model
When M t N , p c, D 1, E 0 (which means supplier as well as retailer offers full
trade credit to his/her customer), let
TRC10 (T )
cD
D M 2 N2
A DTh
D(T M )2
cI p
cI e
T
2
2T
2T
TRC11 (T )
cD
§ T2 N2
·
A DTh
cDI e ¨
(M T ) ¸
¨
¸
T
2
2T
©
¹
(39)
TRC12 (T )
cD
A DTh
cI e D ( M N )
2
T
(40)
(38)
T10*
2 A cD ª M 2 I p I e N 2 I e º
¬
¼
D h cI p
(41)
T11*
2 A cDN 2 I e
D h cI e
(42)
T12*
2A
Dh
(43)
Then TRCci (Ti* ) 0 for i = 10–12. Equation (5a–(5d) will be modified as follows:
TRC(T )
TRC (T ) if M d T
10
°°
®TRC11(T ) if N d T d M
°
°¯TRC12 (T ) if T t N
(44a)
(44b)
(44c)
Equation (44a)–(44c) will be consistent with Equation (5a)–(5d), in Huang (2003),
respectively. Equations (26) and (27) can be modified as ' 2 2 A cI e DN 2
M D h cI e and ' 3
2
2 A N Dh, and hence Theorem 2 reduces to:
2
292
C.K. Jaggi, K.K. Aggarwal and M. Verma
Theorem 5:
1
If ' 2 ! 0 and '3 t 0 then TRC(T * ) TRC(T12* ) and T *
T12* .
2
If ' 2 ! 0 and '3 0, then TRC(T * ) TRC(T11* ) and T *
T11* .
3
If ' 2 d 0 and '3 0, then TRC(T * ) TRC(T10* ) and T *
T10* .
Theorem 5 has been discussed in Theorem 1 of Huang (2003). Hence, Huang (2003) will
be a special case of this paper.
5.3 Goyal (1985) model
When N = 0, which means the supplier would offer the retailer a delay period but the
retailer would not offer the delay period to his/her customer. Therefore, when p c, and
D 1 and E 0. Let
TRC13 (T )
cD
A DTh
D(T M )2
DM 2
cI p
cI e
2
2T
2T
T
(45)
TRC14 (T )
cD
A DTh
ªT
º
cDI e « ( M T ) »
2
2
T
¬
¼
(46)
2 A cDM 2 I p I e
T13*
D h cI p
(47)
2A
D h cI e
T14*
(48)
Then TRCci (Ti* ) 0 for i = 13, 14. Equation (5a)–(5d) will be modified as follows:
TRC(T )
TRC13 (T ) if M d T
®
¯TRC14 (T ) if M t T
(49a)
(49b)
Equation (49a) and (49b) will be consistent with Equation (5a)–(5d), in Goyal (1985),
respectively. Equation (26) can be modified as ' 2 A M 2 D (h cI e ), and Theorem 2
reduces to:
Theorem 6:
1
If ' ! 0 then TRC(T * ) TRC(T11* ) and T *
2
If ' 0, then TRC(T * ) TRC(T10* ) and T *
3
If '
0, then T *
T13*
T14*
T14* .
T13* .
M.
Theorem 6 has been discussed in Theorem 1 of Chung (1998). Hence, Goyal (1985) will
be a special case of this paper.
293
Optimal retailer’s ordering policies
6
Computational analysis
The purposes of this numerical analysis are:
1
to obtain optimal solutions for the two cases
2
to compare the results of present analysis with the previous research work
3
to use sensitivity analysis to highlight the influence of model parameters.
6.1 Numerical analysis
Case 1
M tN
Example 1: To gain more insight into the above theory, we borrowed Huang and Hsu
(2008) example, which is stated as:
A = 80, D = 2000, c = 10, p = 30, h = 7, I p = 0.15, I e = 0.13, M = 0.1, N = 0.08,
E
0.2 and D = 0.09 in appropriate units.
The optimal value of T * and Q* for the present study are T * 0.09454 year and Q* 189
units.
Further, the comparative analysis of proposed model with the previous research work
has been summarised in Table 1.
It is clearly apparent from Table 1 that by implementing partial trade credit policy,
the retailer is able to increase his ordering quantity, which implies that he is able to attract
more customers, which otherwise was not possible, as evident from the Table 1. Hence,
offering partial trade credit at two levels of the supply chain is beneficial for the retailer.
Case 2
M N
Example 2: Let A = 80, D = 5000, c = 10, h =10, I p
and E
0.2, D
0.1, I e
0.2, M = 0.05, N=0.06
0.1 in appropriate units.
The optimal value of T * and Q* for the present model are T *
Q
*
0.057023 year and
285 units.
The sensitivity analysis on different parameters has been presented in Section 6.2.
6.2 Sensitivity analysis
In any decision-making situation, the change in the values of parameters may happen due
to uncertainties. To examine the implications of these changes, the sensitivity analysis
will be of great help in decision-making.
Firstly, we investigate the changing effects of parameters D , p and N for fixed E ,
when M t N in Table 2. Then Table 3 shows the effect of changes in E , p and N for
fixed D for the same case.
From Tables 2 and 3, we have the following inferences:
x
For fixed p and N, an increases in D implies that the retailer’s initial payment to the
supplier decreases, results an increase in the retailer’s order quantity.
294
C.K. Jaggi, K.K. Aggarwal and M. Verma
x
For fixed N and D , when p increases, the optimal cycle length for the retailer
decreases, which in a way increases his inventory turnover. This implies that the
retailer will be ordering more frequently, which eventually helps him to generate
more revenue on the initial payment received from the customer.
x
For fixed p and D , as N increases, i.e. the credit period offered by the retailer to his
customer increases, retailer’s cycle length increases so as his order quantity. This
implies that the retailer is able to attract more and more customers, and he will be
able to generate more revenue on the initial payment received from the customers.
Lastly, for the case M N , Table 4 examines the changes in the parameter D , p and N
for fixed E . It is revealed from Table 4 that for fixed N and ( 1 D ), as p increases,
there is a decrease in the annual total cost.
Table 1
Comparative analysis
T*
Q*
Present study
0.09454
189
Huang (2005)
0.0812
162
0.08648
173
0.08648
173
Research article
N 0, E 0
Huang (2003)
p
c, D
1, E
0
Goyal (1985)
p
c, N
Table 2
1D
0.9
0, D
1, E
0
Optimal solutions when M t N , E fixed
N
p
0.02
10
30
50
10
30
50
10
30
50
10
30
50
10
30
50
0.05
0.08
0.5
0.02
0.05
T*
0.091
0.081
0.074
0.093
0.085
0.079
0.095
0.090
0.087
0.096
0.084
0.076
0.097
0.088
0.081
Q*
TRC (T * ) (in $)
183
164
149
186
170
159
191
182
175
193
170
154
195
176
164
21,489.47
21,204.26
20,898.01
21,513.06
21,282.71
21,039.65
21,556.12
21,421
21,282.18
21,410.46
21,134.23
20,834.35
21,432.99
21,209.92
20,971.89
295
Optimal retailer’s ordering policies
Table 2
1D
0.1
Optimal solutions when M t N , E fixed (continued)
N
p
T*
Q*
0.08
10
0.099
200
21,474.06
30
0.094
188
21,343.3
50
0.090
181
21,207.4
0.02
0.05
0.08
Table 3
10
0.098
197
21,375.43
30
0.086
173
21,103.42
50
0.077
156
20,806.48
10
0.099
199
21,397.49
30
0.089
179
21,177.89
50
0.082
166
20,942.23
10
0.10
204
21,437.71
30
0.09
191
21,309.2
50
0.09
183
21,174.65
Optimal solutions when M t N , D fixed
E
N
0.2
0.02
0.05
0.08
0.4
TRC (T * ) (in $)
0.02
0.05
0.08
p
T*
Q*
TRC (T * ) (in $)
10
0.098
197
21,375.43
30
0.086
173
21,103.42
50
0.077
156
20,806.48
10
0.099
199
21,397.49
30
0.089
179
21,177.89
50
0.082
166
20,942.23
10
0.102
204
21,437.71
30
0.095
191
21,309.2
50
0.091
183
21,174.65
10
0.098
197
21,374.37
30
0.086
172
21,099.8
50
0.077
155
20,799.79
10
0.099
199
21,390.96
30
0.088
177
21,156.03
50
0.081
163
20,902.7
10
0.101
202
21,421.32
30
0.093
187
21,256.3
50
0.088
176
21,082.04
296
C.K. Jaggi, K.K. Aggarwal and M. Verma
Table 3
Optimal solutions when M t N , D fixed (continued)
E
N
0.6
0.02
p
0.05
0.08
Table 4
T*
Q*
TRC (T * ) (in $)
10
0.098
196
21,373.32
30
0.085
172
21,096.18
50
0.077
155
20,793.09
10
0.098
198
21,384.4
30
0.087
175
21,133.92
50
0.080
160
20,862.45
10
0.100
200
21,404.78
30
0.099
182
21,202.1
50
0.084
169
20,985.69
Optimal solutions when M N , E fixed
1D
p
T*
Q*
TRC (T * ) (in $)
0.9
10
0.054988
275
52,885.93
30
0.057023
285.1
52,772.93
50
0.04998
250
52,701.44
0.5
0.1
7
10
0.056389
282
52,807.3
30
0.05859
293
52,691.03
50
0.061072
305.3
52,569.56
10
0.057023
285.1
52,772.93
30
0.059303
296.5
52,655.17
50
0.06188
309.3
52,531.99
Conclusion
This paper extends the assumptions of the two-level trade credit policy in the previously
published results to reflect the realistic situations by incorporating partial trade credit.
It is assumed that the supplier as well as the retailer adopts partial trade credit financing
to stimulate their demand to get optimal inventory policy. The theorems help the retailer
to yield the optimal ordering policy efficiently. Further, we have deduced the models
presented by Huang (2005, 2003) and Goyal (1985). It has been shown that as the
retailer’s initial payment to the supplier decreases, his ordering quantity increases.
Further, for the case M t N , an increase in N results in an increase in his order
quantity, which implies that he is able to attract more customers. Moreover, as the selling
price increases, retailer’s order cycle decreases. This indicates that he should order more
frequently and take the advantage of earning more revenue from the initial payment
received from the customers. The findings have been validated with the help of the
examples. Further, comprehensive sensitive analysis along with a case study has also
been presented.
Optimal retailer’s ordering policies
297
A future study can be incorporated in the present model into more realistic
assumptions such as probabilistic demand, allowable shortages, deteriorating items or
finite replenishment rate.
Acknowledgements
The authors really appreciate the constructive comments by the anonymous referees,
which greatly helped in improving this paper further. The first author would like to
acknowledge the support of Research Grant No. Dean (R)/R&D/2009/487, provided by
University of Delhi, Delhi, India, for conducting this research. The third author would
like to thank University Grant Commission (UGC) for providing the fellowship to
accomplish the research.
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Optimal retailer’s ordering policies
299
Appendix
Case Illustration
A retailer purchases medicine from a pharmaceutical company at c = $100 per unit
and sells it at p = $120 per unit. The company offers a partial credit (i.e. D 0.09 ) for the
time period M = 0.1 year. However, if the payment is not made within the specified
period then 15% interest (i.e. I p 0.15 ) is charged on the outstanding amount. To avoid
default risks, the retailer also offers a partial trade credit (i.e. E 0.02 and N = 0.05 year)
to the credit risk customers. The retailer earns interest at the rate I e 10% on the revenue
generated. Also, the annual demand for the retailer is D = 2,000 units with holding cost
h = $5 per unit per year and the ordering cost A = $250 per order.
Analysis
Using the above proposed model, it is apparent that if both the pharmaceutical company
as well as the retailer offers partial trade credit to its subsequent downstream member,
then the optimal cycle length and order quantity for the retailer is T * 0.1418 year
and Q* 283, respectively. However, when the retailer does not offer any credit period
(i.e. E = 0 and N = 0) to his/her customer, then his cycle length and order quantity
becomes T * 0.1409 year and Q* 281 units, respectively, which implies that the
retailer is able to attract more customers by incorporating partial trade credit, which in
turn increases his order quantity. Thus, partial trade credit is beneficial from the retailer’s
viewpoint.